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Properties

Label 2-966-21.20-c1-0-44
Degree $2$
Conductor $966$
Sign $-0.805 + 0.593i$
Analytic cond. $7.71354$
Root an. cond. $2.77732$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + (−1.18 − 1.26i)3-s − 4-s + 3.87·5-s + (−1.26 + 1.18i)6-s + (0.302 − 2.62i)7-s + i·8-s + (−0.215 + 2.99i)9-s − 3.87i·10-s − 5.48i·11-s + (1.18 + 1.26i)12-s + 4.05i·13-s + (−2.62 − 0.302i)14-s + (−4.57 − 4.91i)15-s + 16-s + 2.53·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.681 − 0.732i)3-s − 0.5·4-s + 1.73·5-s + (−0.517 + 0.481i)6-s + (0.114 − 0.993i)7-s + 0.353i·8-s + (−0.0716 + 0.997i)9-s − 1.22i·10-s − 1.65i·11-s + (0.340 + 0.366i)12-s + 1.12i·13-s + (−0.702 − 0.0809i)14-s + (−1.18 − 1.26i)15-s + 0.250·16-s + 0.615·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 + 0.593i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.805 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.805 + 0.593i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ -0.805 + 0.593i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.499217 - 1.51966i\)
\(L(\frac12)\) \(\approx\) \(0.499217 - 1.51966i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (1.18 + 1.26i)T \)
7 \( 1 + (-0.302 + 2.62i)T \)
23 \( 1 + iT \)
good5 \( 1 - 3.87T + 5T^{2} \)
11 \( 1 + 5.48iT - 11T^{2} \)
13 \( 1 - 4.05iT - 13T^{2} \)
17 \( 1 - 2.53T + 17T^{2} \)
19 \( 1 + 1.59iT - 19T^{2} \)
29 \( 1 + 9.78iT - 29T^{2} \)
31 \( 1 - 6.36iT - 31T^{2} \)
37 \( 1 - 3.09T + 37T^{2} \)
41 \( 1 - 2.14T + 41T^{2} \)
43 \( 1 + 6.76T + 43T^{2} \)
47 \( 1 - 0.573T + 47T^{2} \)
53 \( 1 + 6.95iT - 53T^{2} \)
59 \( 1 + 4.91T + 59T^{2} \)
61 \( 1 - 12.1iT - 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 - 8.75iT - 71T^{2} \)
73 \( 1 - 0.527iT - 73T^{2} \)
79 \( 1 + 1.78T + 79T^{2} \)
83 \( 1 - 0.461T + 83T^{2} \)
89 \( 1 - 12.2T + 89T^{2} \)
97 \( 1 + 8.09iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.961331012208227198979089158388, −9.020463068017103740537848416163, −8.114713879525905791809826542386, −6.89152870988668907145404277023, −6.16235795938862203127385588971, −5.48950094862109245622728304500, −4.41981815628925412725709166391, −2.94170132919848228292285883591, −1.78822259940834574153327398633, −0.869813654858423359533584392396, 1.67424743818792454488967034257, 3.05374669729604161327427031135, 4.69783094880374191082083988569, 5.34189984989617210440840893649, 5.86379580930781371280764775035, 6.63490960080096637486686943467, 7.77272554814747540991110045987, 9.028649644059475834502678548775, 9.549649477547059160086039033926, 10.08920581019378218577576420866

Graph of the $Z$-function along the critical line